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An efficient gradient method using the Yuan steplength

Author

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  • Roberta De Asmundis
  • Daniela di Serafino
  • William Hager
  • Gerardo Toraldo
  • Hongchao Zhang

Abstract

We propose a new gradient method for quadratic programming, named SDC, which alternates some steepest descent (SD) iterates with some gradient iterates that use a constant steplength computed through the Yuan formula. The SDC method exploits the asymptotic spectral behaviour of the Yuan steplength to foster a selective elimination of the components of the gradient along the eigenvectors of the Hessian matrix, i.e., to push the search in subspaces of smaller and smaller dimensions. The new method has global and $$R$$ R -linear convergence. Furthermore, numerical experiments show that it tends to outperform the Dai–Yuan method, which is one of the fastest methods among the gradient ones. In particular, SDC appears superior as the Hessian condition number and the accuracy requirement increase. Finally, if the number of consecutive SD iterates is not too small, the SDC method shows a monotonic behaviour. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Roberta De Asmundis & Daniela di Serafino & William Hager & Gerardo Toraldo & Hongchao Zhang, 2014. "An efficient gradient method using the Yuan steplength," Computational Optimization and Applications, Springer, vol. 59(3), pages 541-563, December.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:3:p:541-563
    DOI: 10.1007/s10589-014-9669-5
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    Citations

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    Cited by:

    1. Marco Viola & Mara Sangiovanni & Gerardo Toraldo & Mario R. Guarracino, 2019. "Semi-supervised generalized eigenvalues classification," Annals of Operations Research, Springer, vol. 276(1), pages 249-266, May.
    2. Behzad Azmi & Karl Kunisch, 2020. "Analysis of the Barzilai-Borwein Step-Sizes for Problems in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 819-844, June.
    3. Yakui Huang & Yu-Hong Dai & Xin-Wei Liu & Hongchao Zhang, 2022. "On the acceleration of the Barzilai–Borwein method," Computational Optimization and Applications, Springer, vol. 81(3), pages 717-740, April.
    4. Corsaro, Stefania & De Simone, Valentina & Marino, Zelda, 2021. "Split Bregman iteration for multi-period mean variance portfolio optimization," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    5. Harry Fernando Oviedo Leon, 2019. "A delayed weighted gradient method for strictly convex quadratic minimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 729-746, December.
    6. Stefania Corsaro & Valentina Simone, 2019. "Adaptive $$l_1$$ l 1 -regularization for short-selling control in portfolio selection," Computational Optimization and Applications, Springer, vol. 72(2), pages 457-478, March.
    7. Yu-Hong Dai & Yakui Huang & Xin-Wei Liu, 2019. "A family of spectral gradient methods for optimization," Computational Optimization and Applications, Springer, vol. 74(1), pages 43-65, September.
    8. Hugo Lara & Rafael Aleixo & Harry Oviedo, 2024. "Delayed Weighted Gradient Method with simultaneous step-sizes for strongly convex optimization," Computational Optimization and Applications, Springer, vol. 89(1), pages 151-182, September.
    9. Masoud Fatemi, 2022. "On initial point selection of the steepest descent algorithm for general quadratic functions," Computational Optimization and Applications, Springer, vol. 82(2), pages 329-360, June.
    10. Crisci, Serena & Ruggiero, Valeria & Zanni, Luca, 2019. "Steplength selection in gradient projection methods for box-constrained quadratic programs," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 312-327.
    11. Serena Crisci & Federica Porta & Valeria Ruggiero & Luca Zanni, 2023. "Hybrid limited memory gradient projection methods for box-constrained optimization problems," Computational Optimization and Applications, Springer, vol. 84(1), pages 151-189, January.
    12. E. Loli Piccolomini & V. L. Coli & E. Morotti & L. Zanni, 2018. "Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm," Computational Optimization and Applications, Springer, vol. 71(1), pages 171-191, September.
    13. Stefania Corsaro & Valentina De Simone & Zelda Marino, 2021. "Fused Lasso approach in portfolio selection," Annals of Operations Research, Springer, vol. 299(1), pages 47-59, April.
    14. Bonettini, Silvia & Prato, Marco & Rebegoldi, Simone, 2016. "A cyclic block coordinate descent method with generalized gradient projections," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 288-300.
    15. Na Huang, 2022. "On R-linear convergence analysis for a class of gradient methods," Computational Optimization and Applications, Springer, vol. 81(1), pages 161-177, January.
    16. Roberto Andreani & Marcos Raydan, 2021. "Properties of the delayed weighted gradient method," Computational Optimization and Applications, Springer, vol. 78(1), pages 167-180, January.
    17. di Serafino, Daniela & Ruggiero, Valeria & Toraldo, Gerardo & Zanni, Luca, 2018. "On the steplength selection in gradient methods for unconstrained optimization," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 176-195.
    18. Clóvis Gonzaga & Ruana Schneider, 2016. "On the steepest descent algorithm for quadratic functions," Computational Optimization and Applications, Springer, vol. 63(2), pages 523-542, March.

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